# The Anti-Magic Square Project

This web site documents my 1999 summer research on a combinatorial design called the Anti-Magic square. Anti-Magic Squares are a variation on the heavily studied and well-understood magic square. In contrast, very little seems to be known about AMSs. These pages describe what was previously known about the structure and history of the AMS and also detail new discoverys regarding their enumeration and construction.

## What is an Anti-Magic Square?

An Anti-Magic Square (AMS) is an arrangement of the numbers 1 to n2 in a square matrix such that the row, column, and diagonal sums form a sequence of consecutive integers. 1

The arrangement to the left is Anti-Magic because sorting the sums (numbers in black on the border) yields the sequence: 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269

This is an example of an AMS(8), or Anti-Magic Square of order 8, which comes from Madachy's Mathematical Recreations

## Purpose

Given this definition, this research project aims to answer some of the following questions:

• What range of sums is it possible to have on the border?
• Is it always possible to construct such a design, or for certain n do they not exist?
• If they do exist for a particular order, how many of them are there?
• Are there any applications of this design to practical problems?

And of course, we aim to have fun doing it! After all this is classified as "Recreational Math!"